We investigate the critical behavior of three-dimensional antiferromagnetic CP(N-1) (ACP(N-1)) models in cubic lattices, which are characterized by a global U(N) symmetry and a local U(1) gauge symmetry. Assuming that critical fluctuations are associated with a staggered gauge-invariant (Hermitian traceless matrix) order parameter, we determine the corresponding Landau-Ginzburg-Wilson (LGW) model. For N=3 this mapping allows us to conclude that the three-component ACP(2) model undergoes a continuous transition that belongs to the O(8) vector universality class, with an effective enlargement of the symmetry at the critical point. This prediction is confirmed by numerical analyses of the finite-size scaling behaviors of the ACP(2) and the O(8) vector models, which show the same universal features at their transitions. We also present a renormalization-group (RG) analysis of the LGW theories for N≥4. We compute perturbative series in two different renormalization schemes and analyze the corresponding RG flow. We do not find stable fixed points that can be associated with continuous transitions.